So You Want to Prove String Theory, Part II: How Can QCD Be a String Theory?

A couple weeks back, I had a post about Nima Arkani-Hamed’s talk at Strings 2016. Nima and his collaborators were trying to find what sorts of scattering amplitudes (formulas that calculate the chance that particles scatter off each other) are allowed in a theory of quantum gravity. Their goal was to show that, with certain assumptions, string theory gives the only consistent answer.

At the time, my old advisor Michael Douglas suggested that I might find Zohar Komargodski’s talk more interesting. Now that I’ve finally gotten around to watching it, I agree. The story is cleaner, more conclusive…and it gives me an excuse to say something else I’ve been meaning to talk about.

Zohar Komargodski has a track record of deriving interesting results that are true not just for the sorts of toy models we like to work with but for realistic theories as well. He’s collaborating with amplitudes miracle-worker Simon Caron-Huot (who I’ve collaborated with recently), Amit Sever (one of the integrability wizards who came up with the POPE program) and Alexander Zhiboedov, whose name seems to show up all over the place. Overall, the team is 100% hot young talent, which tends to be a recipe for success.

While Nima’s calculation focuses on gravity, Zohar and company are asking a broader question. They’re looking at any theory with particles of high spin and nonzero mass. Like Nima, they’re looking at scattering amplitudes, in the limit that the forces involved are weak. Unlike Nima, they’re focusing on a particular limit: rather than trying to fix the full form of the amplitude, they’re interested in how it behaves for extreme, unphysical values for the particles’ momenta. Despite being unphysical, this limit can reveal something about how the theory works.

What they figured out is that, for the sorts of theories they’re looking at, the amplitude has to take a particular form in their unphysical limit. In particular, it takes a form that indicates the presence of strings.

What sort of theories are they looking at? What theories have “particles of high spin and nonzero mass”? Well, some are string theories. Others are Yang-Mills theories … theories similar to QCD.

For the experts, I encourage you to watch Zohar’s talk or read the paper for more detail. It’s a fun story that showcases how very general constraints on scattering amplitudes can translate into quite specific statements.

For the non-experts, though, there’s something that may already be confusing. When I’ve talked about Yang-Mills theories before, I’ve talked about them in terms of particles of spin 1. Where did these “higher spin” particles come from? And where are the strings? How can there be strings in a theory that I’ve described as “similar to QCD”?

If I just stuck to the higher spin particles, things could almost stay familiar. The fundamental particles of Yang-Mills theories have spin 1, but these particles can combine into composite particles, which can have higher spin and higher mass. That should be intuitive: in some sense, it’s just like protons, neutrons, and electrons combining to form atoms.

What about the strings? I’ve actually talked about that before, but I’d like to try out a new analogy. Have you ever heard of Conway’s Game of Life?


Not this one!


This one!

Conway’s Game of Life starts with a grid of black and white squares, and evolves in steps, with each square’s color determined by the color of adjacent squares in the last step. “Fundamentally”, the game is just those rules. In practice, though, structure can emerge: a zoo of self-propagating creatures that dance across the screen.

The strings that can show up in Yang-Mills theories are like this. They aren’t introduced directly in the definition of the theory. Instead, they’re consequences: structures that form when you let the rules evolve and see what they create. They’re another description of the theory, one with its own advantages.

When I tell people I’m a theoretical physicist, they inevitably ask me “Have any of your theories been tested?” They’re operating from one idea of what a theoretical physicist does: propose new theories to describe the world, based on available evidence. Lots of theorists do that, they’re called phenomenologists, but it’s not what I do, or what most theorists I interact with day-to-day do.

So I describe what I do, how I test new mathematical techniques to make particle physics calculations faster. And in general, that’s pretty easy for people to understand. Just as they can imagine people out there testing theories, they can imagine people who work to support the others, making tools to make their work easier. But while that’s what I do, it’s not the best description of what most of my colleagues do.

What most theorists I know do is like finding new animals in Conway’s game of life. They start with theories for which we know the rules: well-tested theories like QCD, or well-studied proposals like string theory. They ask themselves, not how they can change the rules, but what results the rules have. They look for structures, and in doing so find new perspectives, learning to see the animals that live on Conway’s black and white grid. (This is something I’ve gestured at before, but this seems like a cleaner framing.)

Doing that, theorists have seen strings in the structure of QCD-like theories. And now Zohar and collaborators have a clean argument that the structures others have seen should show up, not only there, but in a broader class of theories.

This isn’t about whether the world is fundamentally described by string theory, ten dimensions and all. That’s an entirely different topic. What it is is a question about what sorts of structures emerge when we try to describe the world. What it does is show that strings are, in some sense (and, as for Nima, [with some conditions]) inevitable, that they come out of our rules even if we don’t expect them to.

6 thoughts on “So You Want to Prove String Theory, Part II: How Can QCD Be a String Theory?

  1. Wyrd Smythe

    Do I have this straight that what emerges is not the strings, or structures of strings, but the structures of math that also arise from the analysis of string theories?

    At the least that suggests a similar core, but isn’t that a somewhat of a given for all theories attempting to describe reality? I guess I’m wondering if this is like M-theory in recognizing similarity in disparate approaches?


    1. 4gravitonsandagradstudent Post author

      So, what the actual paper shows is that, in a certain limit, all of these theories have to look like string theory mathematically.

      However, that indicates something more. How a theory behaves in that limit tells us something about the sorts of objects that emerge, mostly what they do when they spin at high energy. And that behavior here is characteristic of strings.

      They may not be the strings of string theory proper, but would still be strings: things with some length and flexibility that can vibrate. (In one of the links I talk about “flux tubes”, areas with lots of gluons between quarks that can stretch out and be “stringy”. That’s one of the ways these sorts of strings could exist.)

      I’m not 100% sure what you mean by “like M-theory”. M-theory is based on dualities, situations where two theories are actually describing the same thing. In that sense, this is similar: the “stringy” description and the “Yang-Mills” description both describing the same behavior. But unlike M-theory, this isn’t about two theories being connected: it isn’t about connecting string theory itself, 10 dimensions and all, to Yang-Mills, rather, it’s about finding that Yang-Mills can contain “stringy” objects.


      1. Wyrd Smythe

        Okay, thanks for the explanation. The flux tube reference suggests an inclusive definition of “stringy” to me when those flux tubes emerge from strong force interaction and consist of gluons whereas strings are thought to be fundamental objects.

        By “like M-theory” I was referring to the unification of the five competing theories that Witten realized all talked about the same thing. My (possibly incorrect) impression is that the dualities you mention still exist within M-theory?


        1. 4gravitonsandagradstudent Post author

          It’s a wider definition, yeah, in that these wouldn’t be fundamental strings. I still wouldn’t call it just a mathematical similarity: they’re still shaped like strings, it’s not just that they happen to use the same equations.

          The dualities I mentioned indeed are within M theory, they’re the links between the five competing theories that showed that they were really about the same thing. This is kind of similar to that, but it’s not really the same story, mostly for the reasons mentioned above.


    1. 4gravitonsandagradstudent Post author

      Yeah, I talked about this a bit in the post on Nima’s talk. They’re definitely following a similar path, but with modern tools they can be more rigorous about it, and show that it works more broadly than I think the S-matrix bootstrap people could tell.



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